 
Summary: Discrete Mathematics 306 (2006) 10681071
www.elsevier.com/locate/disc
Explicit construction of linear sized tolerant networks
N. Alona,b,1
, F.R.K. Chungb
aDepartment of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
bBell Communications Research, 435 South Street, Morristown, New Jersey 07960, USA
Abstract
For every > 0 and every integer m > 0, we construct explicitly graphs with O(m/ ) vertices and maximum degree O(1/ 2),
such that after removing any (1  ) portion of their vertices or edges, the remaining graph still contains a path of length m. This
settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.
© 1988 Published by Elsevier B.V.
1. Introduction
What is the minimum possible number of vertices and edges of a graph G, such that even after removing all but
portion of its vertices or edges, the remaining graph still contains a path of length m? This problem arises naturally
in the study of fault tolerant linear arrays, (see [18]). The vertices of G represent processing elements and its edges
correspond to communication links between these processors. If p, 0 < p < 1 is the failure rate of the processors, it
is desirable that after deleting any p portion of the vertices of G, the remaining part still contains a (simple) path (=
linear array) of length m. Similarly, if , 0 < < 1 denotes the failure rate of the communication links, it is required
that after deleting any portion of the edges of G the remaining part still contains a relatively long path. The objective
